section 9.4 solving equations containing radical expressions
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Section 9.4 Solving Equations Containing Radical Expressions
9.4 Lecture Guide: Solving Equations Containing Radical Expressions
Objective: Solve equations involving radical expressions.
Power Theorem
Algebraically
If x = y, then
.n nx y
Verbally
If two expressions are equal, then their nth powers are equal.
Example
5,x then
2 2
5
25
x
x
2 25 5 but
5 5
Caution: The equations x = y and n nx y
are not always equivalent. The equation n nx y
can have a solution that is not a solution of x = y.
If
For any real numbers x and y and natural number n :
Solving Radical Equations Containing a Single RadicalTo solve equations containing radical expressions, we will use the power theorem, which states that if two expressions are equal, then their nth powers are equal. For example, if x y then 2 2.x y
Procedure Example Step 1. Isolate a radical term on one side of the equation. 5 3 8x
Step 2. Raise both sides to the nth power.Step 3. Solve the resulting equation.*
Step 4. Check the possible solutions in the original equation to determine whether they are really solutions or are extraneous. * If this equation contains a radical, repeat Steps 1 and 2.
Solve each equation.
1. 3 2 0x
Solve each equation.
2. 2 8 4 3x
Solve each equation.
3. 3 3 8 5x
Solve each equation.
4. 2 4 8 7x x
Solve each equation.
5. 3 2 12x x
Solve each equation.
6. 60 36 5x x
Solve each equation.
7.2 3 3x x x
Solve each equation.
8. 7 5x x
9. Use the table and graph to determine the solution of the equation 3 1 1x x
2, 6, 1 by 2, 6, 1
Solution based on table and graph: __________
Solve this equation algebraically.
Determine the exact x- and y-intercepts of the graph of each function.
10. 4 7y x
Determine the exact x- and y-intercepts of the graph of each function.
11. 3 1 2y x
The Pythagorean Theorem: If triangle ABC is a right triangle, then 2 2 2a b c
Use the Pythagorean Theorem to find the length of the side that is not given.
12.
b
2624
Use the Pythagorean Theorem to find the length of the side that is not given.
13.
36
c15
Objective: Calculate the distance between two points.
Distance Formula
The distance d between 1 1,x y and 2 2,x y is given by
2 2.d
Plot each pair of points and then calculate the distance between these points:
14. 5,1 and 3,7
-10
10
-10 10
x
y
Plot each pair of points and then calculate the distance between these points:
15. and
-10
10
-10 10
x
y
1, 6 7,9
Calculate the distance between these points:
16. 14, 10 and 10, 3
Calculate the distance between these points:
17. and 1 1,
6 2
13 3,
6 2
18. Find all points with an x-coordinate of 3 that are 10 units away from the point 5,2 .
-10
10
-10 10
x
yHint: Use the grid below to help.
19. An extension cord is plugged into an outlet at point A on the side of a house. The owner of the house is running an electric weed trimmer to trim along a fence that runs parallel to the side of the house 40 ft away.
(a) Write a function L(x) that gives the length of the extension cord needed to reach any point C along the fence, where x is the distance from B to C and the line segment from A to B is perpendicular to the side of the house.
D
C
B
cord
A
fence
x
40 ft
19. An extension cord is plugged into an outlet at point A on the side of a house. The owner of the house is running an electric weed trimmer to trim along a fence that runs parallel to the side of the house 40 ft away.
(b) Evaluate and interpret L(25).
D
C
B
cord
A
fence
x
40 ft
19. An extension cord is plugged into an outlet at point A on the side of a house. The owner of the house is running an electric weed trimmer to trim along a fence that runs parallel to the side of the house 40 ft away.
(c) If point D is 29 ft from point B, will a 50 ft extension cord allow the owner to reach point D?
D
C
B
cord
A
fence
x
40 ft